Study shows non-compact convex hulls in certain metric spaces.
problem Compactness of convex hulls in weakly non-positive curvature spaces.
method Introduced a conical geodesic bicombing and used it to construct a counterexample.
result Existence of a metric space with a finite subset whose convex hull is not compact.
Study on higher-dimensional quasigeodesics in metric spaces.
problem Understanding asymptotic structure of Morse quasiflats.
method Proving asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity.
result Morse quasiflats exhibit Euclidean volume growth rigidity.
Extends topological results for nonpositive curvature spaces.
problem Characterize and understand topological properties of BNPC spaces.
method Extends results from CAT(0) spaces to BNPC spaces, using links and geodesic bicombings.
result Locally BNPC topological manifolds have discrete singular sets and are homeomorphic to Euclidean space.
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function al…
Stable cubulations and bicombings in mapping class groups and Teichmüller spaces.
problem Understanding geometric structures in mapping class groups and Teichmüller spaces.
method Proving stably approximated by CAT(0) cube complexes, applying to broader colorable hierarchically hyperbolic spaces and groups.
result Stable cubulations and bicombings in mapping class groups and Teichmüller spaces, with stable coarse barycenters.
New bicombings found for mapping class groups and Teichmüller spaces.
problem Finding efficient ways to navigate mapping class groups and Teichmüller spaces.
method Explained bicombings via stable cubical intervals in hierarchically hyperbolic spaces.
result Hierarchical hulls are quasi-isometric to finite CAT(0) cube complexes.
Geodesics and boundaries found for metric structures on hyperbolic groups.
problem Understanding the space of metric structures on hyperbolic groups.
method Outer automorphism invariant geodesic bicombing and boundary construction.
result Boundary contains well-known pseudo metrics and rigidity results.
Let X be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed L-Lipschitz curve γ:S1→X may be extended to an L-Lipschitz map defined on the hemisphere f:H2→X. This implies that X satisfies a quadratic isoperimetri…
Link condition for simplicial complexes to be CUB spaces.
problem Understanding when a simplicial complex is a CUB space.
method Establishing a link condition based on local lattice properties.
result The link condition generalizes Gromov's link condition for cube complexes.
A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bico…
We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bic…
New coarse LS-category introduced for groups and spaces.
problem Large-scale topological properties of groups and spaces.
method Introducing a coarse analog of Lusternik-Schnirelmann category for metric spaces.
result Established lower and upper bounds for geometrically finite and bicombable groups.
The paper proves that certain spaces are injective and Helly graphs.
problem Understanding the structure of certain geometric and algebraic spaces.
method Building Helly graphs and injective metric spaces from lattices.
result The natural piecewise ℓ∞ metric on Euclidean buildings and Deligne complexes is injective. A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.
Study on homogeneous geodesics in sub-Riemannian geometry.
problem Characterizing and understanding homogeneous geodesics in sub-Riemannian manifolds.
method Criterion for geodesics to be homogeneous, proof of geodesic orbit spaces, examples of geodesic orbit sub-Riemannian manifolds.
result Existence of at least one homogeneous geodesic under broad conditions.
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
Study on Mabuchi functional's convexity using ε-geodesics.
problem Understanding the convexity of the Mabuchi functional.
method Analysis of ε-geodesics to study the Mabuchi functional's convexity.
result Uniform fiberwise non-degeneracy of geodesics when Mabuchi functional is ε-affine.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
Characterizes visibility and geodesic loops in complex domains.
problem Visibility and geodesic loops in complex domains.
method Using quasi-geodesic frames to characterize visibility and geodesic loops.
result Characterizes visibility and existence of geodesic loops in Kobayashi complete hyperbolic and Gromov hyperbolic domains.
Conformal geodesics can't spiral in Riemannian manifolds.
problem Existence of spiral conformal geodesics on Riemannian manifolds.
method Analyzing properties of conformal geodesics on Riemannian manifolds.
result No conformal geodesic can become trapped in every neighborhood of a point.
Study geodesics and F-geodesics on tangent bundles over para-Kähler-Norden manifolds.
problem Investigate geodesics and F-geodesics on tangent bundles.
method Investigate geodesics and F-geodesics on tangent bundles and φ-unit tangent bundles equipped with φ-Sasaki metric over para-Kähler-Norden manifolds.
result Investigate and analyze geodesics and F-geodesics on tangent bundles.
New quasi-geodesics for Stiefel manifold simplify complex computations.
problem Efficiently solving geodesic endpoint problem on Stiefel manifold.
method Derived new representations of quasi-geodesics for large-scale computations.
result New quasi-geodesics are closer to Riemannian geodesics.
Study on geodesics of Finsler metrics derived from Riemannian metrics.
problem Investigating geodesics in Finsler metrics derived from Riemannian metrics.
method Proved geodesic lemma for a family of Riemannian geodesic orbit metrics, analyzed geodesic graphs.
result Derived Finsler metrics have geodesic orbit property and belong to a new class of metrics.
Growth rates of geodesics on modular orbifolds are studied.
problem Understanding growth rates of geodesics on modular orbifolds.
method Exhaustion of modular orbifold by compact subsurfaces, analysis of low lying geodesics and reciprocal geodesics.
result Growth rates of low lying geodesics and reciprocal geodesics converge to the full set's growth rate.
We study the geometry of the Thurston metric on Teichmuller space by examining its geodesics and comparing them to Teichmuller geodesics. We show that, similar to a Teichmuller geodesic, the shadow of a Thurston geodesic to the curve graph is a reparametrized quasi-geodesic. However, we show that the set of short curve…
We describe the geometry of geodesics on a Lorentz ellipsoid: give explicit formulas for the first integrals (pseudo-confocal coordinates), curvature, geodesically equivalent Riemannian metric, the invariant area-forms on the time- and space-like geodesics and invariant 1-form on the space of null geodesics. We prove a…
This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.
problem Classifying geodesics of projectively flat sprays and determining sprays based on geodesics.
method Introduction of a geodesic method to determine an n-dimensional spray based on a family of curves with 2(n-1) free parameters as geodesics.
result Classification of geodesics of projectively flat sprays and determination of sprays based on geodesics.
The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.
problem Understanding conditions for geodesic flows to be Anosov and ergodic.
method Analyzing Finsler and Riemannian metrics on surfaces, using recent results.
result Geodesic flows on surfaces are C2 stably ergodic if and only if they are Anosov. Tight geodesics were introduced by Masur-Minsky in [17]. They and their hierarchies have been a powerful tool in the study of the curve complex, mapping class groups, Teichmüller spaces, and hyperbolic 3-manifolds. In the same paper, they showed that there are at least one and at most finitely many tight geodesics betw…
No closed timelike geodesics in Kerr spacetimes, proving absence of closed causal geodesics.
problem Proving the nonexistence of closed timelike geodesics in Kerr spacetimes.
method Analyzing the Kerr-star spacetime, excluding closed null geodesics and proving the nonexistence of closed timelike geodesics.
result No closed timelike geodesics in Kerr spacetimes.
Generic geodesic nets are dense in high-dimensional manifolds.
problem Density of non-closed geodesic nets in high-dimensional manifolds.
method Proving density for a generic metric on a manifold.
result Stationary geodesic nets that are not closed geodesics form a dense set.
Let M be a Margulis spacetime whose associated complete hyperbolic surface S has compact convex core. Generalizing the correspondence between closed geodesics on M and closed geodesics on S, we establish an orbit equivalence between recurrent spacelike geodesics on M and recurrent geodesics on S. In contrast, no timeli…
The authors define a class of functions on Riemannian manifolds, which is called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming i…
Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.
problem Counting ambiguous geodesics in curved spaces.
method Asymptotic formula for common perpendiculars in negatively curved spaces, applying to modular orbifolds and number fields.
result Confirms and extends Motohashi's conjecture on binary additive divisor problem.
Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.
problem Behavior of geodesics on cones over arbitrary Riemannian manifolds.
method Show existence of first integrals uniquely determining geodesics.
result Geodesic flow on cones is superintegrable and Liouville--Arnold integrable for non-radial trajectories.
The study examines geodesics and tight geodesics in surface curve complexes.
problem Characterizing the spectrum of geodesics and tight geodesics in curve complexes.
method Analyzing the number of geodesics and tight geodesics of length d in curve complexes. result The spectrum of geodesics is a subset of the spectrum of tight geodesics, with equality for geodesics of length 2.
Theorem shows generic metrics yield non-degenerate geodesic nets.
problem Characterizing geodesic nets on generic metrics.
method Proving all connected embedded nets are non-degenerate for Baire-generic metrics.
result All stationary geodesic nets are non-degenerate for generic metrics.
New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…
Study shows how lengths of geodesic arcs determine linking number of Legendrian knots.
problem Linking number of Legendrian knots on negatively curved surfaces.
method Analyzes Poincaré series on negatively curved surfaces.
result Explicit rational value of Poincaré series at 0 interprets linking number of Legendrian knots.
Study on geodesics in Kropina metrics with applications.
problem Existence of connecting and closed geodesics in Kropina metrics.
method Analytical proofs and applications to null geodesics and navigation problems.
result Proves existence of geodesics in Kropina metrics.
Study inverse problems for twisted geodesic flows on manifolds.
problem Understanding inverse problems for twisted geodesic flows.
method Generalized ray transforms and tensor tomography.
result New insights into rigidity problems for twisted geodesic flows.
There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…
Self-crossing geodesics on convex surfaces are studied.
problem Understanding patterns of geodesics crossing themselves.
method Analyzing closed geodesics on convex surfaces.
result Self-crossing geodesics exist on convex surfaces.
We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. …
In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogen…
Homoclinic orbits found in geodesic flows on surfaces.
problem Existence of homoclinic orbits in geodesic flows.
method Kupka-Smale metric on closed surfaces.
result Homoclinic orbits for all hyperbolic geodesics.
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
The study proves geodesic loops and chords without intersections for specific metrics.
problem Existence of geodesic loops and chords without self-intersections.
method Genericity results, perturbation, and existence theorems.
result For specific metrics, geodesic loops and chords exist without intersections.